Crystallography of homophase twisted bilayers: coincidence, union lattices and space groups

A general scheme is proposed to classify and determine the crystallographic properties of twisted bilayers of any homophase 2D structures using complex numbers.


Introduction
The discovery of strong electronic correlations and superconductivity in twisted bilayer graphene (Trambly de Laissardiè re et al., 2010Laissardiè re et al., , 2012, with a so-called magic rotation angle close to 1.05 where the Fermi velocity vanishes, has significantly increased the interest in detailed study (Cao, Fatemi, Demir et al., 2018; of these kinds of low-dimension structures [see, for transition metal dichalcogenides, Naik & Jain (2018), Wu et al. (2019), Soriano & Lado (2020), Venkateswarlu et al. (2020)]. The eventual aim is to determine which symmetry property may explain the existence of flat bands in the electronic structure (Suarez Morell et al., 2010): what, in the symmetry properties (if any) of twisted bilayers, is at the origin of this electronic localization?
A robust answer to this question requires a practical and simple crystallographic description of bilayer structures. This is the focus of the present work. The fundamental mathematical aspects of coincidence lattices at any dimensions are to be found in the very elaborated studies of Pleasants et al. (1996), Baake & Grimm (2006), Baake & Zeiner (2017). We focus here on the very elementary practical aspect of investigating the unique case of 2D bilayer structures.
Investigation of the symmetry properties of the superimposition of two 3D crystals, called bicrystals, was carried out in the 1980s (Gratias & Portier, 1982;Pond & Vlachavas, 1983) in the study of the properties of grain boundaries in metals and alloys. Although, at that time, these bicrystals were only theoretical concepts, their 2D versions of superimposing two monoatomic layers make sense in the present context as the idealization of a twisted bilayer considered as the superimposition of two infinitely thin monoatomic layers differently oriented by a twist rotation of angle perpendicular to the layer plane and displaced with respect to each other by a translation in the plane.
The paper is organized as follows. Our first task is to enumerate which specific rotation angles lead to a situation where two homophase layers share a common sublattice, say T , of index AE in Ã, and to explicitly give the expressions of these sublattices T and those U generated by the union of the lattices of the two layers. Our second task is to understand how these specific coincidence angles are distributed with respect to the values of the square length (identical to AE for the square and hexagonal systems) of the coincidence unit-cell vectors. Our third task is to determine which space group N is generated for bilayers with coincidence lattices when the rigid-body translation varies at constant rotation . Three appendices give the explicit illustration of the whole process in the case of twisted graphene bilayers and the conditions for coincidence and union lattices to exist in the case of heterophase bilayers obtained by dilatation and/or rotation or mechanical deformation. We use the following notation: point groups are noted in capital letters like G or W; space groups and translation groups are noted in calligraphic letters like G or T ; space symmetry operators (or functions in the complex plane as discussed next) are noted as b a a or b g g whereas point symmetry operators are simply written as or g.

Elementary bicrystallography
As already mentioned, homophase bilayers are ideally defined here as the superimposition of two identical monolayers on top of each other, forming an infinitely thin layer of matter. The twist operation, that transforms the monolayer I into II, is either a rotation-translation b a a acting as b a a r = ðjÞr = r þ , or a mirror translation (in all 2D enantiomorphic structures, these two descriptions are equivalent as they describe the same twist operation) b m m ; oriented along a direction of angle with the x axis, acting as b m m ; r = ðm jÞr = m r þ . The original monolayer I has space group [we use the notations of Hahn (2005)] G with point group À and lattice Ã showing the holohedral symmetry class of point group G Ã with G Ã ! À according to: (i) Oblique system G Ã ¼ 2: À ¼ 1; 2; (ii) Rectangular system G Ã ¼ 2mm: À ¼ m; 2mm; (iii) Square system G Ã ¼ 4mm: À ¼ 4; 4mm; (iv) Hexagonal system G Ã ¼ 6mm: À ¼ 3; 3m; 6; 6mm. The corresponding group and lattice of the second monolayer II are given by ; À 0 ¼ À À1 and Ã 0 ¼ Ã: Since any point in the orbit of r under G can be equivalently chosen, we characterize the transformation from layer I to II by the set b a a G. The inverse transformation from II to I is given by G b a a À1 ¼ Gð À1 j À À1 Þ as shown in Fig. 1.

Using complex numbers for 2D crystallography
2D crystallography is particularly simple to handle using complex numbers. In fact, any 2D vector V ¼ ðx; yÞ in an orthonormal reference frame of the plane is equivalently described by a complex number z ¼ x þ iy 2 C. Concerning the nodes of a 2D lattice Ãða; bÞ defined by its unit cell of vectors a and b, we choose the unit-cell vector a along the real axis and its length as the length unit with no loss of generality. Passing from monolayer I to monolayer II is achieved by the set b a a G and from II to I by the inverse set Gb a a À1 . The overlap of the monolayers, designed here as a bilayer, generates its own symmetry that is a 2D space group if the two lattices Ã and Ã 0 have a common coincidence lattice T ¼ Ã \ Ã 0 and only a quasiperiodic symmetry otherwise.

Figure 2
The lattice Ãða; bÞ with unit cell ða; bÞ is the set of complex numbers z ¼ n þ m expði'Þ, n; m 2 Z, 2 R, where the unit vector a is chosen as the unity along the real axis x and b is the complex number b ¼ expði'Þ. A rotation of angle around the origin transforms z into z expðiÞ and a mirror m passing through the origin and oriented along the direction of angle transforms z into " z z expð2iÞ.
the real axis as shown in Fig. 2. A general primitive lattice Ã p of unit vectors a ¼ 1 and b ¼ expði'Þ is then the set of complex numbers [In addition, c-type lattices encountered in the rectangular symmetry class The complex notations of the 2D lattices are given in Table 1.
The symmetry operations act as functions of complex variable f ðzÞ as elementary transformations of complex numbers: (i) A translation u 2 C acts on a point z as z ! uðzÞ ¼ z þ u; (ii) A rotation around the origin transforms z into z ! ðzÞ ¼ z expðiÞ; (iii) A mirror m passing through the origin and oriented in the direction transforms z into z ! m ðzÞ ¼ " z z expð2iÞ. Space operators are the usual combinations of point symmetries and translations as shown in Table 2.

Coincidence angles for homophase bilayers
(This includes bilayers with different monolayers but sharing identical lattices.) General twisted bilayers are quasiperiodic structures built on a Z module of rank 4. Specific cases arise for particular values of the rotation angle , called coincidence angles, where the two initial lattices Ã and Ã 0 share a 2D sublattice T , called the coincidence lattice characterized by the index AE [defined by equation (9)], the ratio of the unit-cell sizes of T and ÃðÃ 0 Þ. This makes the nodes of the general Z module of rank 4 condense on a 2D lattice U called the union lattice, discussed later, in a similar way to generating periodic approximants from quasicrystals. In fact, as will be shown next, coincidence angles occur an infinite countable number of times and form a uniformly dense set of values on the real axis: any generic twisted bilayer is infinitely close to a coincidence situation which is the only case leading to exact space symmetries of the bilayer.
Finding the proper coincidence angles has been the subject of a very large number of publications for 2D and 3D crystals (see, for instance, Ranganathan, 1966;Grimmer, 1973Grimmer, , 1974Grimmer, , 1984. The most complete and recent analysis of coincidence lattices in 2D crystals has been given by Romeu et al. (2012), a work that we reconsider here briefly using complex notations and that leads to a derivation which is simple and gives explicit expressions for the coincidence and union (homophase) lattices, as discussed next.
Let be the rotation angle from the first monolayer to the second, both of point group G. The coincidence lattice, if any, is the common subset of the lattice translations of the monolayers: A first necessary condition for a coincidence lattice to possibly exist is that a lattice row defined by the node ðn; mÞ with n; m 2 Z; gcd ðn; mÞ ¼ 1 superimposes on another one ðn 0 ; m 0 Þ of the same orbit under G by the rotation around the origin: The possible generic solutions are listed below according to the crystalline system of the structure. {There are a few specific cases, in particular for the square system, with n 2 þ m 2 = n 02 þ m 02 where the nodes ðn; mÞ and ðn 0 ; m 0 Þ do not belong to the same orbit under G [for instance the nodes (3, 4) and (5, 0)]. These cases are not explicitly considered here.} (i) Oblique system of point group 2: the generic orbit contains only two terms ðx; yÞ; ð" x x; " y yÞ, so that there are no solutions but the trivial rotation ¼ .
(ii) Rectangular system {this includes the special case of those specific oblique lattices where 2 cos ¼ AE1 which should be considered as c-type rectangular lattices [cmðmÞ]} point group 2mm 0 ð' ¼ =2Þ: the generic orbit contains four terms ðx; yÞ; ð" x x; yÞ; ðx; " y yÞ; ð" x x; " y yÞ; the two non-trivial solutions are those where ðn 0 ; m 0 Þ is deduced from ðn; mÞ by the mirrors m x ðn 0 ¼ Àn; m 0 ¼ mÞ and m 0 y ðn 0 ¼ n; m 0 ¼ ÀmÞ. (iii) Square system of point group 4mm 0 ð' ¼ =2; ¼ 1Þ: in addition to the rectangle case, new solutions are n 0 ¼ AEm; m 0 ¼ AEn. All these solutions can be generated by using the mirror m x and the mirror m 0 xy rotated by =4 up to additional =2 rotations.
(iv) Hexagonal system 6mm 0 ð' ¼ 2=3; ¼ 1Þ: here too, all possible solutions are obtained by using the mirror m x and the mirror m 0 rotated from the x axis by =6 up to additional rotations of =3.
Hence, with the exception of the oblique system which presents no generic solutions, the rotation of a lattice node on top of one of its equivalents can be achieved using the two research papers Acta Cryst. (2023). A79, 301-317 Gratias and Quiquandon Crystallography of twisted bilayers 303 Table 1 2D lattices: the parameter a is the length unit ða ¼ 1Þ along the real axis.
All lattices are primitive except in the rectangular system with c-type lattices. Running indices n; m are integers. Here, j ¼ expð2i=3Þ.

System
Lattice Ãðn; mÞ Table 2 2D symmetry operations ðgjtÞ acting as functions f ðzÞ in the complex plane z ¼ x þ iy.
mirror generators of these point groups (see, for instance, Coxeter, 1963) according to: rectangle (2mm 0 ) mirrors in the directions ¼ 0 and =2; square (4mm 0 ) mirrors in the directions ¼ 0 and =4; hexagonal (6mm 0 ) mirrors in the directions ¼ 0 and =6. Let ¼ 2 be the rotation around the origin that superimposes the node z 0 ¼ " z z expð2iÞ, on top of z = n þ m expði'Þ related to z 0 by the mirror m oriented along the direction as shown in Fig. 3. Putting ¼ z" z z, we note that ffiffiffi p expðiÞ = z expðÀiÞ and therefore ffiffiffi p expðiÞ = z expðÀiÞ so that and thus These three relations apply for the rectangle, square and hexagonal systems with the following specific forms: These relations are necessary conditions for ensuring two equivalent lattice rows superimpose on each other by the rotation ¼ 2 and ¼ 2ð À Þ. Because these two solutions differ only by the constant rotation , we consider from now on the unique solution ¼ 0 defined by the basic relations remembering that with each solution ð0 Þ is associated the solution 0 ðÀ 0 ¼ À 0Þ.
2.2.1. Coincidence lattices in the rectangle system.
Ensuring one row in coincidence is of course not sufficient to generate a 2D coincidence lattice: this requires another noncollinear row of lattice nodes to be in coincidence for the same rotation angle. We discard the oblique system that we know has no generic rows of coincidence whatever the rotation angle and thus no possible coincidence lattice. We focus now on the unique rectangle system since the square and hexagonal systems are specific high-symmetry cases of the rectangle one.
Let fT 1 ; T 2 g be the unit cell of the coincidence lattice T we seek with T 1 ¼ n þ im in the rectangle system. Because T , if it exists, shares at least the same symmetry class as the lattice of the monolayer (see, for instance, Gratias & Portier, 1982)here the rectangular symmetry 2mm or higher -another coincidence vector T 2 ¼ n 0 þ im 0 exists that is aligned along iT 1 up to a certain ratio r: This requires rn ¼ m 0 , thus r 2 Q and Àmr 2 ¼ n 0 which is achieved if and only if 2 2 Q, i.e. 2 ¼ p=q, where p and q are coprime positive integers. Thus, is a rational number: These results confirm in a few calculation steps those obtained by Romeu et al. (2012) following a seminal paper by Ranganathan (1966) in the context of classical 3D crystallography. Here, coincidence lattices in homophase bilayers in the rectangular system exist if and only if the ratio ¼ jbj=jaj is the square root of a rational number: ; p; q; 2 N: [Oblique lattices with 2 cos ' ¼ 1 as the hexagonal lattice can be considered as rectangular c-type lattices of parameters ð1; 0 ¼ tan 'Þ and therefore show a 2D coincidence lattice when 02 ¼ tan 2 ' ¼ p=q, with p; q 2 Z.] We conclude therefore that the coincidence angles for the rectangle system are For all 2D systems except the oblique case, which has no generic solution, two homophase layers share the same crystallographic row defined by the lattice node z ¼ n þ m expði'Þ, with n and m coprimes, if there exists an equivalent lattice node z 0 ¼ " z z expð2iÞ deduced from z by a mirror in the direction defined by: distributed as a uniformly dense countable set of points on the real axis as ¼ 2 arctan with 2 Q.
2.2.2. Explicit expression of the coincidence lattice in the rectangle system. The unit vector T 2 of T is the smallest vector along iT 1 with integer coordinates It is obtained by multiplying iT 1 by q and then dividing the result by ¼ gcdðmp; nqÞ: We first note that putting nq ¼ r and mp ¼ s with r; s 2 Z, we obtain T 2 ¼ Às þ ir, which explicitly shows that, indeed, T 2 belongs to Ã. We then observe that, as required, T 2 is orthogonal to T 1 , but the length of T 2 is in the ratio with the length of T 1 only when q ¼ and therefore, although with at least the same symmetry class as Ã, the coincidence lattice is not necessarily homothetic to Ã in the general case as illustrated in Fig. 4. Because of relations (7), we have so that the coincidence lattice T is explicitly given by showing that the coincidence lattice T is generated by a lattice characterized by a 0 ¼ 1 and b 0 ¼ iðq=Þ, rotated by with respect to Ã and linearly dilated by ffiffiffi p . Since AE is the index of the translation group T in Ã, i.e. the ratio of the surfaces of the unit cell of the coincidence lattice T with respect to the one of the lattice Ã, we find which is, indeed, an integer since ¼ gcdðmp; nqÞ is a divisor of n 2 q þ m 2 p. 2.2.3. The union lattice. The other fundamental translation group is the group U generated by the union of the lattice translation groups of the two crystals: This shows that, for any coincidence angle and any symmetry class larger than or equal to the rectangular one, U is homothetic to T in the linear ratio 1=AE (this ratio applies on each unit vector leading thus to a relative density of nodes card U =T ¼ AE 2 ). It is easily demonstrated that this relation holds for the square and hexagonal (see Appendix A) systems with the coincidence lattices given by Example of coincidence lattices in black in the rectangular system

Figure 5
We visualize the set of coincidence angles in plotting the values of the coincidence angles 0 ¼ =2 ¼ arctan m=n as a function of the square length ¼ n 2 þ m 2 2 of the generating coincidence node ðn; mÞ.
Here, the example of the rectangle system with ¼ ffiffiffiffiffiffiffi 5=3 p for 0 =2; the other solution 0 ¼ À =2 is obtained on the same diagram by exploring the x axis from À=2 to 0. The vertical dashed line

Coincidence patterns P ¼ ð; Þ
A classical scheme in metallurgy consists of collecting all the possible coincidence angles , each associated with its AE index, in a general pattern P of points ð; AEÞ which is the superimposition of all the coincidence angles equivalent to with respect to the intrinsic symmetries of the layer, each associated with its AE. In the case of a rectangle system, this pattern can exhibit quite a complicated fine structure due to the arithmetic irregularities introduced by the term q= in the definition of AE seen in equation (9). Moreover, this kind of pattern is heavily redundant because of the superimposition of several rotations that are equivalent with respect to the inner symmetry of the layer. In fact, as shown in Fig. 5, a simpler and equally informative pattern is obtained by plotting only one rotation representative in the elementary sector of the point group of the monolayer, as a function of the square length of the superposition node ðn; mÞ instead of AE: A very basic fact is that since the coincidence angles are defined by lattice vectors ðn; mÞ, where n and m are coprime integers, these vectors point to those nodes of a 2D lattice known as the set of points visible from the origin, noted here V, as shown in Fig. 6. All points ð; Þ of the coincidence pattern P are in a one-to-one correspondence with those ðn; mÞ, of V.
In particular, rational rows in the set V faithfully mirror the branches in P that are asymptotically converging to specific angles characterized by their coincidence nodes ðn; mÞ with tanðÞ ¼ m=n as exemplified by the rows and corresponding branches drawn in cyan and purple in Figs. 6 and 7.
The simplest way to classify and order these branches is to label them according to Farey sequences f ðNÞ (see, for instance, Hardy & Wright, 1979). The Farey sequence of order N, noted f ðNÞ, is the set of fractions m=n where m and n are coprime integers, associated with the nodes ðn; mÞ of the set V [see, for instance, in a different context Philippon (2008)], and such that 0 < m < n N, ordered by size.
We note the following properties: (i) For any two elements of a sequence, corresponding in the set V to the nodes ðn 0 ; m 0 Þ pointing in the direction tanðm 0 =n 0 Þ and ðn 1 ; m 1 Þ pointing in the direction tanðm 1 =n 1 Þ, with m 0 =n 0 < m 1 =n 1 , the vector ðn 0 þ n 1 ; m 0 þ m 1 Þ pointing along their diagonal is such that (ii) If two elements i and j are consecutive (j ¼ i þ 1) in a sequence with m i =n i < m j =n j then n i m j À m i n j ¼ 1. Because of Bezout's identity, we deduce that beyond ðn i ; m i Þ and ðn j ; m j Þ being coprimes, the pairs ðn i ; n j Þ and ðm i ; m j Þ are also coprimes. The points of coincidence are defined by coprime pairs of integers ðn; mÞ, i.e. by fractions m=n in their irreducible forms. Plotted on the nodes of a lattice, they generate the so-called set V of points visible from the origin made of the lattice points drawn in blue. Each rational row of this set, as those drawn in colors, is associated with a branch of points with the same colors in the coincidence pattern of Fig. 7. Distribution of the coincidence angles tanðÞ ¼ m=n n; m > 0 versus ¼ n 2 þ m 2 2 in logarithmic scale, for the rectangle lattice with . The points are distributed on branches asymptotically converging to specific coincidence angles ¼ arctanðm i =n i Þ where the points ðn i ; m i Þ belong to (extended) Farey sequences generated from the initial pair ½ð1; 0Þ; ð0; 1Þ; here, the optimum branches generated by the Farey sequence F r ð3Þ asymptotic by lower (purple) and upper (cyan) values are underlined with the same colors as their corresponding rows on the set V of Fig. 6.
In fact, because the coincidence angles run between 0 and for the rectangle system, the sequences we are interested in here are extended Farey sequences (Halphn, 1877), noted F r ðNÞ, made of the standard Farey sequence f ðNÞ between (1, 0) and (1, 1) completed by the sequence from (1, 1) to (0, 1), obtained in adding to the original sequence the inverse fractions n=m in opposite order. Such typical extended sequences for the rectangular system, where 0 =2, are

Invariance property of the branches
Defining branches of points in the coincidence pattern is pertinent when the points of the same branch, described by a running index k, share the same property independent of this index. To determine which invariance property a branch corresponds to, we note that, because of relations (1), (2) and (3), any two points ð; Þ, associated with the coincidence node ðn; mÞ, and ð 0 ; 0 Þ, associated with ðn 0 ; m 0 Þ, of the same coincidence pattern are related by where Á ; 0 ¼ À 0 . Relation (13) is the key for characterizing the invariance rule for each branch of the pattern.
We consider the set of nodes under their irreducible form [gcdðn k;k 0 ; m k;k 0 Þ ¼ 1], defining the points in P: As shown in Fig. 6, at constant k 0 and running k, these nodes ðn k;k 0 ; m k;k 0 Þ describe rows in V that are parallel to the direction ðn 0 ; m 0 Þ. At constant k and running k 0 , they describe rows in the direction ðn 1 ; m 1 Þ. These two rows intersect at the node ðn 0 þ n 1 ; m 0 þ m 1 Þ.
Observing that we note that at constant k 0 and running k, the points ðn k;k 0 ; m k;k 0 Þ describe a set of branches in P, one for each value of k 0 , asymptotic (by upper values for k 0 > 0 and by lower values for k 0 < 0) to 0 for k ! 1 where all points share the invariance property: Similarly, from relation (17), at constant k and running k 0 , corresponds a set of branches asymptotic (by upper values for k > 0 and by lower values for k < 0) to 1 for k 0 ! 1 sharing the invariance property 8 k 0 2 Z; ffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 k;k 0 p sin Á 1;ðk;k 0 Þ ¼ k: Concerning the irreducibility property, we note that n k;k 0 and m k;k 0 are both multiples of gcdðk; k 0 Þ and therefore k and k 0 must be coprime for the node ðn k;k 0 ; m k;k 0 Þ to belong to V. Thus, any row in the set generated by a running kðk 0 Þ at constant k 0 ðkÞ exhibits only the points that are not multiples of the prime factors of the constant k 0 ðkÞ. For example, in the Farey sequence Fð0Þ ¼ ½ð1; 0Þ; ð0; 1Þ where ðn k;k 0 ; m k;k 0 Þ = ðk; k 0 Þ, the rows parallel to the x corresponding to running k at constant k 0 show, in increasing k 0 order: all k values for k 0 ¼ 1, only odd values of k for k 0 ¼ 2, k not a multiple of 3 for k 0 ¼ 3, k not a multiple of 2 and 3 for k 0 ¼ 6 etc. The densest rows correspond to k 0 being a prime number. The same behavior is to be found for the rows parallel to the y direction and, extraordinarily enough, for any row parallel to a rational direction.
The kðk 0 Þ branches associated with the smallest values of k;k 0 , designated here as optimal branches because they generate the smallest coincidence unit cell, are those where the constant kðk 0 Þ in relations (16) and (17) is the unity. These are the branches and associated rows colored, respectively, in cyan and purple in Figs. 6 and 7.
The two optimal branches in k and k 0 defined by the neighbor nodes ðn 0 ; m 0 Þ, ðn 1 ; m 1 Þ in the Farey sequence intersect at the node defined by k ¼ k 0 ¼ 1, i.e. at the node ðn 0 þ n 1 ; m 0 þ m 1 Þ which is precisely the term inserted between the two original nodes in the Farey sequence next to the original one.

Analytical expression of the optimal branches
Although the coincidence angles form a dense enumerable set of points on the trigonometric circle, the proximity of two alpha values does not ensure that of their corresponding values. This happens only when the two angles are on the same branch. Two branches are particularly important which are asymptotic to the angles of the generating mirrors of the point symmetry of the lattice of the monolayer, i.e. ¼ 0 for all systems, with additional ¼ =2 for a rectangle, ¼ =4 for a square and ¼ =6 for a hexagonal system. They have a particular importance for bilayers with very small rotations as they allow us to choose the smallest-sized coincidence lattices closest to the angle we seek generating the smallest atomic model to be used in electronic calculations.
In the rectangle system, the two extreme asymptotic angles are ¼ 0 and =2 associated with the two extreme branches defined by the Farey sequence [ðn 0 ; m 0 Þ = (1, 0), ðn 1 ; m 1 Þ = (0, 1)]. The relation (4) leads to p : ( We assume > 1 (p > q). Using 2 ¼ p=q with AE ¼ q= and ¼ gcdðmp; nqÞ, we find AE k;1 ¼ q gcdðp; kÞ k;1 ; AE 1;k 0 ¼ q gcdðq; k 0 Þ 1;k 0 : We first observe, as shown in Fig. 8, that each time k and p share the same divisor, AE k;1 changes its value so that the initially unique k;1 branch splits into subbranches AE k;1 , where is the number of divisors of p. [Let n ¼ a p 1 Â b p 2 Â c p 3 Â . . . be the positive integer of prime factors a, b, c, . . . ; the number of its divisors is ðp 1 þ 1Þ Â ðp 2 þ 1Þ Â ðp 3 þ 1Þ Â . . ..] Similarly, 1;k 0 splits into subbranches AE 1;k 0 , where is the number of divisors of q. We then note that the same angle is shared by the two branches at steps, respectively, k and k 0 when kq ¼ pk 0 , i.e. for k ¼ p'; k 0 ¼ 1 for one branch and k 0 ¼ q'; k ¼ 1, for the other, ' 2 Z. At that stage q p';1 ¼ p 1;q' and therefore AE p';1 = ðq=pÞ p';1 = 1;q' ¼ AE 1;q' . The two branches superimpose every p steps for one branch and q steps for the other with the same AE values. Hence, the optimal branch for small angles in the rectangle system ( 2 ¼ p=q; p > q) is found to be with a primitive lattice of parameters A = 1 þ iq', B = Àp' þ i that condenses to a c-type lattice ðA þ BÞ=2, ðA À BÞ=2 when p, q and ' are all three simultaneously odd. For the square system, the situation is much simpler since here ¼ AE. The Farey sequence to be used here is ½ð1; 0Þ; ð1; 1Þ out of which we obtain The same angle is shared by the two branches each time k ¼ 2k 0 with, then, k;1 ¼ 2 1;k 0 . This is easily understood by noting that k ¼ 2k 0 implies n k;1 þ m k;1 ¼ 2 þ 2k 0 and n k;1 À m k;1 ¼ 2k 0 both even, which lead directly to n 1;k 0 = 1 þ k 0 , m 1;k 0 = k 0 with a value twice smaller. The branch 308 Gratias  Splitting of the optimal branches as plotted against AE instead of in the rectangle system for the cases ¼ ffiffiffiffiffiffiffiffiffi . The red dots and lines correspond to AE k;1 and the blue dots and lines to AE 1;k 0 . In the first case, p ¼ 2 Â 5 the red branch splits into four subbranches in the ratio 1, 2, 5 and 10, and since q ¼ 7, the blue branch splits into two subbranches of ratio 1 and 7. In the second case, p ¼ 21 ¼ 3 Â 7, the red branch splits into four subbranches in the ratio 1, 3, 7 and 21, whereas for q ¼ 2 4 , the branch splits into five subbranches in the ratio 1, 2, 4, 8 and 16. In both cases, the red and blue branches superimpose for k ¼ p' and k 0 ¼ q'; ' 2 Z. asymptotic to ¼ =4 is therefore the optimal solution with the smallest unit cell defined by with a primitive lattice defined by The case of the hexagonal system is treated in Appendix A and leads to These results are easily understood by noting that the smallest coincidence angles are obtained when the two superimposed nodes are as close as possible to each other. Indeed, applying relation (4) to the rectangle system with For the square system with ¼ =4 and n ¼ ' þ 1, m ¼ ', relation (4) gives and for the hexagonal system, with ¼ =6 and n ¼ 2' þ 1, m ¼ '

Space groups of homophase bilayers with coincidence lattices
Building the space group of a homophase bilayer with a coincidence lattice is a simple work in principle that follows the same general scheme: the symmetry group of a set of two identical objects taken as a whole is the union of the symmetry elements that are common to both objects and are intrinsic symmetries of these objects plus extra elements, if any, that exchange the two objects as illustrated in Fig. 9. It is easily demonstrated that the union of these two sets forms a group (see, for instance, Gratias & Quiquandon, 2020). Here, both the rotation and the rigid-body translations are to be considered in the computations of these two basic sets: (i) The intersection group contains those symmetry elements of the original layers that are of the same nature and superimpose in space, ; this group is never empty since it contains at least the identity and the translation group T of the coincidence lattice.
(ii) The additional set of symmetry elements correspond to those extra new elements that exchange the two layers, transforming layer I into II and simultaneously II into I, defined by the intersection of the cosets b a a G with G b a a À1 designated, if not empty, the exchange set E: The symmetry group of the homophase bilayer, say N of translation group T , is thus the union In addition to the group N , another fundamental symmetry group of interest is the space group W which generates, for a given rotation and a given rigid-body translation , all translations 0 that generate equivalent bilayers, i.e. bilayers that can superimpose on top of each other by an isometry. Any two such translations and 0 for the same rotation are said to be equivalent. They form the orbit of in W .
Because the rigid-body translation acts as a global translation of the layers, it is sufficient to consider only the orientational symmetry, i.e. the point group À, instead of the whole space group G. The point group W is obtained as in the preceding case, by considering the intersection of the point groups I and II and the exchange set but with the major change that, since the elements of the exchange set transform into its opposite, we must multiply the exchange set by the inversion operation: where " 1 1 stands for the inversion operator. [A very unfortunate mistake is to be corrected in the work of Gratias & Quiquandon (2020) where the inversion operation has been forgotten in the expression of W and improperly added in the one of N .] Here again, it is easily shown that W is a group. These two identical cups share the same mirror (blue frame) and transform into each other by another mirror (red frame) perpendicular to the previous one. Alone, each cup has the point symmetry m, but the pair of cups, taken as a whole, has point symmetry 2mm.
The translation subgroup of W is found owing to the fact that adding to any translation of the lattices of either crystal transforms the bilayer into one of its equivalents. This translation group is the union group U introduced in Section 2.2.3. The set of equivalent translations to is therefore the orbit of in the space group W ðU Þ generated by the product of W with the translation group U : Hence, the number of different symmetry groups of the bilayers induced by varying the rigid-body translation for a given rotation is the number of strata of the group W (U ). For example, as shown in Appendix A and Fig. 13, there are only six different space groups for graphene bilayers whatever the rigid-body translation (and whatever the rotation). It also shows that the natural reference frame to be used for labeling is the union lattice. Moreover, the domain of definition of the rigid-body translation becomes very narrow and decreases linearly as 1= ffiffiffiffi AE p for coincidence angles tending towards zero. As a consequence, at very small angles of rotation where the coincidence lattice unit cell increases dramatically, the unit cell of the union lattice becomes small enough for the rigid-body translation to become physically meaningless. Therefore, in the case of large moiré patterns due to small disorientations, it is not necessary to consider the rigid-body translation in the description (it can be chosen to be the null vector).

Finding the bilayer groups: point symmetry
The point symmetry elements to consider are rotations z ! z expðiÞ and mirrors z ! " z z expð2iÞ with ; 2 {0, 6 , 4 , 3 , 2 }. The rotation commutes with all the rotations of the lattice crystal since z ! z expðiÞ expðiÞ ¼ z exp½ið þ Þ ¼ z expðiÞ expðiÞ: The intersection point group À \ À À1 is thus the set of all the rotations of the point group of the crystal whatever the value of the coincidence angle .
On the other hand, the exchange set À \ À À1 contains the mirrors generated by the product of the rotation by the original mirrors, i.e. mirrors rotated by ¼ =2 from the original ones. Indeed, the elements of À act on z as z ! expðiÞ" z z expð2iÞ = " z z exp½ið þ 2Þ whereas those of À À1 act as z ! z expðÀiÞ expð2iÞ = " z z exp½ið þ 2Þ. Therefore the exchange sets contain all the mirrors obtained by a rotation of ¼ =2 of the original mirrors of the structure whatever the value of the coincidence angle . This explains why the coincidence lattice T has the same point symmetry as the original symmetry class of the lattice. For the group W , the exchange set that contains all the mirrors is multiplied by the inversion, generating thus an equivalent set of mirrors but rotated by =2 with respect to the initial ones.

Finding the bilayer groups: space symmetry
The space group W is easily determined since it is the direct product of W with U .
Concerning the group N , the calculation requires a few steps.
Elements of I are the elements b g g;b g g 0 2 G such that b a ab g g = b g g 0 b a a and elements of E are the elements b a ab g g such that b a ab g g = b g g 0 b a a À1 .
From b a az = z expðiÞ þ and b a a À1 z = z expðÀiÞ À expðÀiÞ, with b g g being either a rotation b g gz = z expðiÞ þ t, or a mirror b g gz = " z z expð2iÞ þ t, we have the general explicit expressions b a ab g gz ¼ For an element being possibly included in either set I or E, the arguments of the variable z must be identical for the equalities to hold for any value of z.
Concerning I, the comparison between the lines (28) and (29) shows that the only possible solution for the elements b g g and b g g 0 to be in I is two rotations of the same angle such that t 0 À t expðiÞ ¼ À expðiÞ: Concerning E, the comparison of (28) with (30) shows that the pertinent elementsb a ab g g are obtained withb g g andb g g 0 being parallel mirrors such that with t; t 0 2 U or possibly U =2 for the c-type space groups cm and c2mm and the non-symmorphic ones pg, p2mg, p2gg and p4gm.
Since t 0 À t expðiÞ is a vector of U or U =2, we find that: (i) The rotation ðjtÞ 2 G is in I if is such that À expðiÞ is a vector of U (or U =2) which is achieved for pointing to special positions of the group W .
(ii) The mirror ðm þ jtÞ is in E if is such that þ " exp½2ið þ Þ is a vector of U or U =2 requiring thus to point along the perpendicular bisector of a mirror of E and thus to align along a mirror of W .
These two conditions lead to non-trivial solutions for being located at special positions of W ðU Þ.
This shows that the space group N depends on the value of according to the different symmetry strata of W ðU Þ: the number of different possible space groups of the bilayer is equal to the number of symmetry strata of the group W ðU Þ.
Moreover, the type of space group N of the bilayer does not depend on the value of the coincidence angle : whatever the value of in that set is, the groups N obtained for rigidbody translations with the same coordinates in U are isosymbolic; their actual representations in space are scaled according to the length ffiffiffi p and the rotation ¼ =2.

A simple low-symmetry example
We consider two bilayers A and B with coincidence lattices built from structures of symmetry class m x with space groups pm x ðÃÞ for A and pg x ðÃÞ for B. In both cases, the point group W is made of the identity for the intersection group and m y (original m x rotated by ¼ =2 plus =2 because of the inversion) for the exchange set: W ¼ pm y with translation group U . This group has three strata expressed in the unit cell of U : ð0; yÞ, ð1=2; yÞ and ðx; yÞ with little groups, respectively, m y , m y and 1. The translation ¼ ð0; yÞ expressed in the unit cell of U generates the group N ¼ pm x for the structure A and N ¼ pg x for structure B, both of translation group T , and vice versa for the translation ¼ ð1=2; yÞ (Fig. 10).

Conclusion
To summarize, we find that infinitely many coincidence lattices generically exist down to the rectangle symmetry provided that the ratio of the lengths of the unit-cell vectors is the square root of a rational number: 2 ¼ p=q. They are generated by specific coincidence rotations of angle of the form ¼ 2 ¼ 2 arctanðm=nÞ where n and m are coprime integers and can be written as where ¼ n 2 þ 2 m 2 and ¼ gcdðmp; nqÞ.
With each coincidence lattice is associated a union lattice U ¼ T =AE homothetic to T and which is the translation group of the space group of the equivalent translations of the rigid-body translation . Both coincidence and union lattices share at least the symmetry class of the original layer. For square and hexagonal systems, the three lattices T , Ã and U are two-bytwo homothetic in the linear ratio ffiffiffiffi AE p . The complete set of possible coincidence lattices characterized by the rotation angle and the unit-cell size AE of the corresponding coincidence lattice form a diagram in one-toone correspondence with the so-called set of points visible from the origin and can be analyzed using Farey sequences. They are distributed on branches, each characterized by a geometric invariant relating the sinus of the rotation angle to the square root of the unit-cell size.
In the case where a coincidence lattice exists, the space group of the bilayer depends on the value of the rigid-body translation between the two layers. There are as many different symbolic names of space groups as there are strata in the group W ðU Þ of the equivalent translations to a given one. These symbolic names do not depend on the value of the rotation .
Because the group W ðU Þ has U as translation subgroup, the unit-cell size of which tends to zero for rotations tending to zero, the rigid-body translation becomes a non-pertinent parameter -analogous to a phason field in quasicrystals -for twisted bilayers with very small rotation.  , n = 1, m = 1, = 101.537 ; the coincidence lattice T is defined by T 1 ¼ ð1; 1Þ; T 2 ¼ ð " 3 3; 2Þ with AE ¼ 5.
A subsequent work will discuss the case of general bilayers where U is a Z module of rank 4 in connection with the notion of 0-lattice which is independent of the possible existence of a coincidence lattice.

APPENDIX A The specific example of graphene bilayers
Graphene has a 2D periodic structure of group p6mmða; bÞ with a carbon atom at special position 2b 3m ð1=3; 2=3Þ. It is described in the complex plane C by the primitive hexagonal lattice Ã ( ¼ 1; ' ¼ 2=3) defined by Ã ¼ fz ¼ n þ jm; j ¼ expð2i=3Þ; n; m 2 Zg with a carbon atom at position z 1 ¼ ð1 þ 2jÞ=3 [and equivalently z 2 ¼ ð2 þ jÞ=3] as shown in Fig. 11(a). [It turns out that the commonly used notation in the physics community of graphene is to take the hexagonal reference frame with the acute angle =3 instead of the crystallographic definition that uses the angle 2=3. This corresponds to choosing ðÀ " j j; 1Þ as reference frame instead of ð1; jÞ. Noting thus that a node z can be equivalently written as z ¼ n þ mj ¼ M À N " j j we obtain n ¼ N þ M and m ¼ N and thus M ¼ n À m.] The unit-cell parameter equal to a = 0.2456 nm is chosen here as the unit length.
The point symmetry elements of 6mm are generated by the rotation of =3 located at the origin and transforming (the six hexagonal rotations are f1; À " j j; j; À1; " j j; Àjg) z into À " j jz and the mirror along x transforming z into " z z. The orbit G z of a generic point z has thus 12 elements per unit cell: G z ¼ fz; À " j jz; jz; Àz; " j jz; Àjz; " z z; Àj" z z; " j j" z z; À" z z; j" z z; À " j j" z zg þ Ã as exemplified in Fig. 11(b).

A1. Graphene bilayers with coincidence lattices
Twisted graphene bilayers are certainly among the most studied materials in the world [see, for instance, the recent review by Geim (2009)], often created, for example, in epitaxial graphene growth on the C-terminated face of Si-C (see Campanera et al., 2007;Hass et al., 2008;Varchon et al., 2008;Bistritzer & MacDonald, 2011). These twisted bilayers are the superimposition of two single graphene sheets slightly twisted with respect to each other by a small angle of a few degrees or less. It was seen a couple of years ago that these twisted graphene bilayers have remarkable electronic structures (see Trambly de Laissardiè re et al., 2010). As already mentioned, from the geometric point of view discussed here, the two graphene sheets are considered as infinitely thin and located on the same plane.
Twist rotations of angle leading to coincidence lattices are infinitely many (see Feuerbacher, 2021). They are characterized by the rotations that superpose a representative of a given orbit of nodes z ¼ n þ jm on top of another point of the same orbit G z . As previously mentioned, because of the high symmetry of the hexagonal system, it is enough to examine the rotation ¼ 2 around the origin that transforms the lattice point z ¼ n þ " j jm ¼ n À m À jm into z 0 ¼ n þ jm as shown in Fig. 11, where n and m are positive coprime integers with n > 2m > 0: and thus ffiffiffiffi AE p expðiÞ ¼ n þ jm; where ¼ =2 and AE ¼ n 2 þ m 2 À nm, as application of equation (7)  (a) Graphene is a 2D structure made of a honeycomb lattice of carbon atoms (in black on the picture). The standard primitive hexagonal lattice is generated by the pair 1; j ¼ expð2i=3Þ in complex notations defining the unit cell drawn in gray. The point symmetry group is 6m which can be generated by the two mirrors M and M 0 . (b) Generating coincidence lattices by rotation is easily obtained by applying rotations around the origin that superpose nodes deduced from each other by the mirror along the real axis transforming the node n þ " j jm into n þ jm. Because of the hexagonal symmetry, choosing point n þ " j jm with n; m coprime in the region n > 2m > 0 between the mirrors M and M 0 is sufficient for generating all the possible rotations of coincidence angles and =3 À .

Figure 12
The rotations of coincidence n þ " j jm ! n þ jm are defined by ¼ arctan m ffiffi ffi 3 p =ð2n À mÞ n; m 2 Z, n > 2m > 0 plotted for one unique twist rotation as a function of AE ¼ n 2 þ m 2 À nm for AE < 8000, on a logarithmic scale. As in the general case, well defined asymptotic branches are observed which correspond to the terms of the consecutive Farey sequences: the asymptotic branches of the F h ð6Þ sequence are drawn in cyan and red.
The same angle is found between the two branches for k ¼ 3k 0 with k;1 ¼ 3 1;k 0 . Indeed, since then n k;1 = 2 þ 3k 0 we see that n k;1 þ m k;1 = 3ð1 þ k 0 Þ and, of course, n k;1 À 2m k;1 = 3k 0 are both multiples of 3, leading to a unit cell three times smaller. This shows that the asymptotic branch ! =6 is the optimal solution leading to the smallest unit cells:

A5. Symmetry groups of twisted graphene bilayers
Our last task is to analyze the overall symmetry of the graphene twisted bilayers with coincidence lattices. The group W ðU Þ is easily found as W ðU Þ ¼ p6mm of translation group U whatever the value of the coincidence angle in P. The group p6mm contains six symmetry strata listed in Table  3. There are thus only six different possible space groups for the bilayer according to the coordinates of expressed in units of the union lattice U as shown in Table 3 and exemplified in Fig. 13.
The case of very small rotations deserves some attention. Rotations decreasing to zero are associated with coincidence lattices with larger and larger unit cells and therefore to shorter unit cells for the union lattices. As noticed in the body of the text, this leads to rigid-body translations tending to zero and therefore losing any physical pertinence. Small rotations can indeed be locally described as translations between the two almost-parallel layers, as shown in Fig. 14(a). The normalizer at ¼ 0 is the group of the graphene layer, p6mm, scaled by ffiffiffiffi AE p and rotated by : the bilayer has exactly the same symmetry properties as the initial graphene unit cell but magnified to mesoscopic scales. It can be roughly described as Example of a graphene bilayer with twist rotation ðn; mÞ ¼ ð3; 1Þ, ¼ 2 arctan ffiffi ffi 3 p =5 = 38.2132 , T 1 ¼ ðÀ1; À2Þ; T 2 ¼ ð2; 3Þ, AE = 7, as a function of the rigid-body translation shown in Table 3. All coincidence angles in P generate symmetry groups with the same symbols; they differ only by the scaling factor defined by the union and coincidence lattices. Table 3 Symmetry groups N of graphene twisted bilayers as a function of the rigid-body translation expressed on the basis of the union group W ðU Þ ¼ p6mm 0 with origin chosen on the sixfold axis.
Its translation group is the union lattice U defined by U 1 ¼ ðn þ jmÞ=AE and U 2 ¼ ½Àm þ jðn À mÞ=AE with AE ¼ n 2 þ m 2 À nm. an hexagonal tiling made of three main microscopic highsymmetry structures occurring at each special point of the large hexagonal coincidence unit. In that renormalization-like view, the centers of the initial hexagons are replaced by the structure of local symmetry 6mm, usually designated as AA, that is the graphene itself, the carbon atoms by the structure of local symmetry 3m, called ABðBAÞ, as in the natural graphite, and the binding between carbon atoms, designated as SP, by the structure 2mm shown in Fig. 14(b) of space group c2m with cell parameters ða ¼ 1; b ¼ ffiffi ffi 3 p Þ and two carbon atoms at positions ð0; 1=3Þ and ð1=2; 1=3Þ. These three basic highsymmetry structures, 6mm, 3m and 2mm, correspond to the first three special positions of dimension 0 as shown in Table 3 and Fig. 13. This exhausts all the symmetry possibilities: there are no other kinds of structures, for any ( 1 and any (here, meaningless) values of .

APPENDIX B
Heterophase bilayers with coincidence lattices B1. Homogeneous dilatation-rotation coincidence lattices for heterophase bilayers Heterophase bilayers are formed by two layers of different structures; they show very similar geometrical properties to the homophase bilayers. However, our present context of using complex numbers allows us to treat here only those heterophase bilayers where the lattices Ã and Ã 0 of the layers can be deduced from each other by a dilatation-rotation, i.e. when Here, % is the dilatation coefficient and the rotation from Ã to Ã 0 . This kind of transformation is the general case when the lattices belong both to either the square or the hexagonal systems as examplified in Fig. 15.
We choose here to discuss heterophase bilayers in the square system for simplicity.